The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton

Transcription

1 Journl of Modrn hysics, 014, 5, ublishd Onlin August 014 in SciRs. htt:// htt://dx.doi.org/.436/jm Th Angulr Momnt Diol Momnts nd Gyromgntic Rtios of th Elctron nd th roton A. Gorgiou School of hysics Astronomy nd Mthmtics, Univrsity of Hrtfordshir, Htfild, UK Rcivd 3 Jun 014; rvisd 1 July 014; cctd 4 July 014 Coyright 014 by uthor nd Scintific Rsrch ublishing Inc. This work is licnsd undr th Crtiv Commons Attribution Intrntionl Licns (CC BY). htt://crtivcommons.org/licnss/by/4.0/ Abstrct W hd rviously obtind nlyticl formul for th diol momnts nd ngulr momnt of rotting shricl bodis. Th rsulting formul wr lid to th Sun, th str 78 Virginis nd th Erth. Th grmnt of th thorticl formul with th ctul rl situtions is indd rmrkbl. In this not w ly th sm formul to th lctron nd th roton, using th clssicl vlus of th rdii, so no untum mchnicl trtmnt is considrd. Kywords Angulr Momnt, Diol Momnts, Elctron, roton 1. Introduction Th uros of this work is to obtin nlyticl formul for th diol momnts nd ngulr momnt of th lctron nd th roton. or this uros, xct solutions of th Einstin-Mxwll fild utions for volum distributions of rotting chrgd mttr r ruird. Thr r difficultis in obtining such solutions, but ths wr succssfully trtd in dtil in [1]. In ddition, thr r uncrtintis rgrding th rdii of ths rticls, rsulting in som diffrncs btwn th clcultd numricl vlus nd th cctd vlus of th diol momnts.. Th Elctron Th mss m, clssicl rdius, th lctric chrg of th lctron nd th vcuum sd of light, r: How to cit this r: Gorgiou, A. (014) Th Angulr Momnt Diol Momnts nd Gyromgntic Rtios of th Elctron nd th roton. Journl of Modrn hysics, 5, htt://dx.doi.org/.436/jm

2 m g cm = su c m. Th following rtios r ruird: m= s = = G c = (1) () If th mss is m, thn thr is n lctric chrg m G ssocitd with m [1]. Thus, if thr is n dditionl lctric chrg, th totl chrg is + m G su. Th vlu of m G is only nd so it is ngligibl comrd to th vlu of in (1). If =, w my thrfor writ for th chrg nd its sur, = = = = 19 In th Rissnr-Nordstrom solution [], th cofficint of with givn by th scond of (3). Th cofficint of dt in th Krr-Nwmn mtric, is with Eution (5) thn, bcoms RN dt in th mtric, hs th form givn by Gm G = 1 + (4) 4 cr cr j sin θ = r + j cos θ = j + r RN. (6) Gmr G c c = r + j cos θ whr j is th ngulr momntum r unit mss; s Eutions (18)-(0) of []. On θ = π, cosθ = 0 nd so on r = Eution (7) bcoms = 1 Gm G 4 c + c (8) which is th sm s Eution () of [] (Not tht in rfrnc [] f is usd whr w us ). It follows tht instd of Eutions (7)-(8) of [1], w hv for r = Gm c λ = (9) 4 In ccordnc with th rsults of [1], th diol momnt, totl ngulr momntum J nd gyromgntic rtio r: G c (3) (5) (7) 155

3 whr Th vlus in () nd (3) giv for Eutions () thn giv Th cctd vlu is ( 1 λ ) 3 = bλ 3 3 c J = b λ G G = b( 1 λ ) J c ( ) rom th scond of (), w obtin for J Th gyromgntic rtio is thrfor, c G () 1 Gm G b = C( ) = 1+. (11) 4 C c c λ in (9) nd λ λ = λ = = rg/guss, or (1) = J/T. (13) 4 = J/T. (14) J 8. (15) J = (16) nd this is rcisly th vlu of G c. It must b notd tht thr r vrious diffrnt vlus for th rdius of th clssicl lctron. Th vlu of found is smllr thn th cctd vlu nd it dnds on which rdius w choos. Thus, if w us rdius with numricl vlu = 0 cm (17) which is 7 tims lrgr thn th clssicl rdius of th lctron nd rt th clcultions, w find 4 = J/T. (18) This is nrr th cctd vlu in (14). It ws stblishd by Dhmlt [3] tht th ur limit for th lctron rdius is cm. This simly imlis tht if is th lctronic rdius, w must hv <. Th vlus of th rdii w usd, including th clssicl rdius of th lctron, r much lrgr thn this. 3. Th roton Th mss, rdius, lctric chrg nd mss to rdius rtio of th roton, r rsctivly: m g 0.84 cm. m su (19) If thr is n dditionl lctric chrg, th totl chrg will b + m G su. Th vlu of m G is 8 only su nd so it is ngligibl comrd to th vlu of in (1). If =, w my 156

4 thrfor writ for th totl chrg nd its sur, = = = = Th sm ormul (1) for λ nd () for, J nd J r vlid, but with th roton rmtrs in Eutions (19) instd of th lctron ons. W find λ = λ = = J = J = = W not tht th gyromgntic rtio found in th 5 th of Eutions (1), is rcisly th vlu of G c. In th cs of th roton th vlu of found is lrgr thn th cctd vlu, which is 4. Conclusion (0) (1) J/T W hv obtind th diol momnts ngulr momnt nd gyromgntic rtios of th lctron nd th roton using th nlyticl formul dvlod in [1]. In th cs of th lctron, th vlu of found is smllr thn th cctd vlu, but in th roton cs it is lrgr. If Dhmlt s [3] dductions r vlid, comlt rvlution is ncssry using our nlyticl ormul () to find, J nd J. It is not ossibl to do this, bcus Dhmlt, dos not giv ny dfinit vlus for th rdii; h only stts tht th lctron rdius should stisfy th inulity <. But in ny cs, th uros of th clcultions hr, is to s if th vlus of th known clssicl lctron nd roton rdii, giv th xctd vlus of, J nd J. Rfrncs [1] Gorgiou, A. (01) Journl of Modrn hysics, 3, [] Blindr, S.M. (003) Dirc s Elctron vi Gnrl Rltivity. ACS No: D, C. [3] Dhmlt, H. (1988) hysicl Scrit, T, -1. htt://dx.doi.org/.88/ /1988/t/

5